Substructural Logics

Substructural logics are non-classical logics weaker
than classical logic, notable for the absence of structural
rules present in classical logic. These logics are motivated by
considerations from philosophy (relevant logics), linguistics (the
Lambek calculus) and computing (linear logic). In addition, techniques
from substructural logics are useful in the study of traditional
logics such as classical and intuitionistic logic. This article
provides a brief overview of the field of substructural logic. For a
more detailed introduction, complete with theorems, proofs and
examples, the reader can consult the books and articles in the
Bibliography.

Logic is about logical consequence. As a result, the
conditional is a central notion in logic because of its
intimate connection with logical consequence. This connection is
neatly expressed in residuation condition:

p, q
⊢
r if and only if p
⊢
q →
r

It says that r follows from p together with
q just when q → r follows
from p alone. The validity of the transition from q
to r (given p) is recorded by the conditional
q → r.

This connection between the conditional and consequence is called
residuation by analogy with the case in mathematics. Consider
the connection between addition and substraction. a +
b = c if and only if a =
c − b. The resulting a
(which is c − b) is the
residual, what is left of c when b is taken
away. Another name for this connection is the deduction
theorem.

However, there the connection between consequence and the
conditional contains one extra factor. Not only is there the turnstile,
for logical consequence, and the conditional, encoding consequence
inside the language of propositions, there is also the comma,
indicating the combination of premises. We have read
“p, q
⊢
r” as “r follows from ptogether withq”. To combine premises is to
have a way to take them together. But how can we take them
together? It turns out that there are different ways to do so, and so,
different substructural logics. The behaviour of premise combination
varies as the behaviour of the conditional varies. In this introduction
we will consider some examples of this.

1.1 Weakening

It is one thing for p to be true. It is another for the
conditional q → p to be true. Yet, if
‘→’ is a material conditional, q →
p follows from p. For many different reasons, we may
wish to understand how a conditional might work in the absence
of this inference. This is tied to the behaviour of premise
combination, as can be shown by this demonstration.

p ⊢ p

p, q ⊢ p

p ⊢ q → p

From the axiomatic p
⊢
p (anything follows from itself) we deduce that
p follows from p together with q, and then
by residuation, p
⊢
q → p. If we wish to reject the inference from
p to q → p, then we either reject
residuation, or reject the identity axiom at the start of the proof, or
we reject the first step of the proof. It is illuminating to consider
what is involved in this last option. Here, we are to deny that
p follows from p, q. In general, we are to
reject the inference rule that has this form:

X ⊢ A

X, Y ⊢ A

This is called the rule of weakening. The rule steps from a
stronger statement, that A follows from X to a
possibly weaker one, that A follows from X together
with Y.

People have offered different reasons for rejecting the rule of
weakening, depending on the interpretation of consequence and premise
combination. One of the early motivating examples comes from a concern
for relevance. If the logic is relevant (if to say that
p entails q is true is to say, at least, that
q truly depends on p) then the comma need
not not satisfy weakening. We may indeed have A following
from X, without A following from
X,Y, for it need not be the case that A
depends on X and Y together.

In relevant logics the rule of weakening fails on the other
side too, in that we wish this argument to be invalid too:

q ⊢ q

p, q ⊢ q

p ⊢ q → q

Again, q may follow from q, but this does not mean
that it follows from ptogether withq,
provided that “together with” is meant in an appropriately
strong sense. So, in relevant logics, the inference from an arbitrary
premise to a logical truth such as q → q may
well fail.

1.2 Commutativity

If the mode of premise combination is commutative (if anything which
follows from X, Y also follows from Y,
X) then we can reason as follows, using just the identity
axiom and residuation:

p → q ⊢ p → q

p → q, p ⊢ q

p, p → q ⊢ q

p ⊢ (p →q) → q

In the absence of commutativity of premise combination, this proof is
not available. This is another simple example of the connection between
the behaviour of premise combination and that of deductions involving
the conditional.

There are many kinds of conditional for which this inference fails.
If “→” has modal force (if it expresses a
kind of entailment, in which p → q is true when
in every related circumstance in which p holds, q
does too), and if
“⊢”
expresses local consequence
(p⊢q
if and only if any model, at any circumstance at which p
holds, so does q) it fails. It may be true that Greg is a
logician (p) and it is true that Greg’s being a
logician entails Greg’s being a philosopher (p →
q – in related circunstances in which Greg is a
logician, he is a philosopher) but this does not entail that
Greg is a philosopher. (There are many circumstances in which the
entailment (p → q) is true but q is
not.) So we a circumstance in which p is true but (p
→ q) → q is not. The argument is
invalid.

This counterexample can also be understood in terms of behaviour of
premise combination. Here when we say X,A
⊢
B is true, we are not just saying that B
holds in any circumstance in which X and A both hold.
If we are after a genuine entailmentA → B, then
we want B to be true in any (related) circumstance in which
A is true. So, X,A
⊢
B says that in any possibility in which A is
true, so is B. These possibilities might not satisfy all of
X. (In classical theories of entailment, the possibilities are
those in which all that is taken as necessary in X
are true.)

If premise combination is not commutative, then residuation can go
in two ways. In addition to the residuation condition giving
the behaviour of →, we may wish to define a new arrow ← as
follows:

p, q
⊢
r if and only if q
⊢
r ←
p

For the left-to-right arrow we have modus ponens in this
direction:

p → q, p
⊢
q

For the right-to-left arrow, modus ponens is provable with the
premises in the opposite order:

p, q ← p
⊢
q

This is a characteristic of substructural logics. When we pay attention
to what happens when we don't have the full complement of structural
rules, then new possibilities open up. We uncover two
conditionals underneath what was previously one (in intuitionistic or
classical logic).

In the next section we will see
another example
motivating non-commutative premise combination and these two different
conditionals.

1.3 Associativity

Here is another way that structural rules influence proof. The
associativity of premise combination provides the following proof:

p → q, p ⊢ qr → p, r ⊢ p

p → q, (r → p, r) ⊢ q

(p → q, r → p), r ⊢ q

p → q, r → p ⊢ r → q

p → q ⊢ (r → p) → (r → q)

This proof uses the cut rule at the topmost step. The idea is
that inferences can be combined. If X
⊢
A and Y(A)
⊢
B (where
Y(A) is a structure of premises possibly including
A one or more times) then Y(X)
⊢
B too (where
Y(X) is that structure of premises with those
instances of A replaced by X). In this proof, we
replace the p in
p → q, p ⊢ q
by
r → p, r on the basis of
the validity of
r → p, r ⊢ p.

1.4 Contraction

A final important example is the rule of contraction which
dictates how premises may be reused. Contraction is crucial in the
inference of p → q from p →
(p → q)

p → (p → q) ⊢ p → (p → q)

p → q ⊢ p → q

p → (p → q), p ⊢ p →q

p → q, p ⊢ q

(p
→ (p → q), p),
p
⊢
q

p
→ (p → q), p
⊢
q

p
→ (p → q)
⊢
p → q

These different examples give you a taste of what can be done by
structural rules. Not only do structural rules influence the
conditional, but they also have their effects on other connectives,
such as conjunction and disjunction (as we shall see below) and
negation (Dunn 1993; Restall 2000).

1.5 Structure on the right of the turnstile

Since the introduction of Gentzen’s sequent calculus (Gentzen
1935), we have known that the difference between classical
logic and intuitionistic logic can also be understood as a
difference of structural rules. Instead of considering sequents of the
form X
⊢
A, in
which we have a collection of antecedents and a single consequent, for
classical logic it is fruitful to consider sequents of the form

X
⊢
Y

where both X and Y are collections of statements. The
intended interpretation is that from all of the X it
follows that some of the Y. In other words, we cannot
have all of the X and none of the Y obtaining.

Allowing sequents with multiple consequents and translating the
rules into this expanded context, we are able to derive classical
tautologies. For example, the derivation

p ⊢p

p ⊢ q, p

⊢ p → q, p

shows that either p → q or p must hold.
This is classically valid (if p fails, p is
false, and conditionals with false antecedents are true), but
invalid in intuitionistic logic. The difference between classical and
intuitionistic logic, then, can be understood formally as a difference
between the kinds of structural rules permitted, and the kinds of
structures appropriate to use in the analysis of logical consequence.

There are many different formal systems in the family of substructural
logics. These logics can be motivated in different ways.

2.1 Relevant Logics

Many people have wanted to give an account of logical validity which
pays some attention to conditions of relevance. If
X,A
⊢
B holds,
then X must somehow be relevant to A.
Premise combination is restricted in the following way. We may have
X
⊢
A without
also having X,Y
⊢
A . The new material Y might not be relevant to the
deduction. In the 1950s, Moh (1950), Church (1951) and Ackermann (1956)
all gave accounts of what a ‘relevant’ logic could be. The
ideas have been developed by a stream of workers centred around
Anderson and Belnap, their students Dunn and Meyer, and many others.
The canonical references for the area are Anderson, Belnap and
Dunn’s two-volume Entailment (1975 and 1992). Other
introductions can be found in Read’s Relevant Logic,
Dunn and Restall’s “Relevance Logic” (2002), and
Mares’ Relevant Logic: a philosophical
interpretation (2004).

2.2 Resource Consciousness

This is not the only way to restrict premise combination. Girard (1987)
introduced linear logic as a model for processes and resource
use. The idea in this account of deduction is that resources must be
used (so premise combination satisfies the relevance criterion) and
they do not extend indefinitely. Premises cannot be
re-used. So, I might have X,X
⊢
A, which says that I can use X twice
to get A. I might not have X
⊢
A, which says that I can use X once
alone to get A. A helpful introduction to linear logic is
given in Troelstra’s Lectures on Linear Logic (1992).
There are other formal logics in which the contraction rule
(from X,X
⊢
A to
X
⊢
A) is absent.
Most famous among these are Łukasiewicz’s many-valued
logics. There has been a sustained interest in logics without this rule
because of Curry's paradox (Curry 1977, Geach 1995; see also
Restall 1994 in Other Internet Resources).

Independently of either of these traditions, Joachim Lambek considered
mathematical models of language and syntax (Lambek 1958, 1961). The
idea here is that premise combination corresponds to composition of
strings or other linguistic units. Here X,X differs
in content from X, but in addition, X,Y
differs from Y,X. Not only does the number of
premises used count but so does their order. Good
introductions to the Lambek calculus (also called categorial
grammar) can be found in books by Moortgat (1988) and Morrill
(1994).

We have already seen a fragment of one way to present substructural
logics, in terms of proofs. We have used the residuation condition,
which can be understood as including two rules for the conditional, one
to introduce a conditional

X, A ⊢ B

X ⊢ A →B

and another to eliminate it.

X ⊢A → BY ⊢ A

X, Y ⊢B

Rules like these form the cornerstone of a natural deduction system,
and these systems are available for the wide sweep of substructural
logics. But proof theory can be done in other ways. Gentzen
systems operate not by introducing and eliminating connectives, but by
introducing them both on the left and the right of the turnstile of
logical consequence. We keep the introduction rule above, and replace
the elimination rule by one introducing the conditional on the left:

X ⊢ AY(B) ⊢ C

Y(A → B, X) ⊢ C

This rule is more complex, but it has the same effect as the arrow
elimination rule: It says that if X suffices for A,
and if you use B (in some context Y) to prove
C then you could just as well have used A →
B together with X (in that same context Y)
to prove C, since A → B together with
X gives you B.

Gentzen systems, with their introduction rules on the left and the
right, have very special properties which are useful in studying
logics. Since connectives are always introduced in a proof
(read from top to bottom) proofs never lose structure. If a
connective does not appear in the conclusion of a proof, it will not
appear in the proof at all, since connectives cannot be eliminated.

In certain substructural logics, such as linear logic and the Lambek
calculus, and in the fragment of the relevant logic R
without disjunction, a Gentzen system can be used to show that the
logic is decidable, in that an algorithm can be found to
determine whether or not an argument X
⊢
A is valid. This is done by searching for
proofs of X
⊢
A
in a Gentzen system. Since premises of this conclusion must feature no
language not in this conclusion, and they have no greater complexity
(in these systems), there are only a finite number of possible
premises. An algorithm can check if these satisfy the rules of the
system, and proceed to look for premises for these, or to quit if we
hit an axiom. In this way, decidability of some substructural logics is
assured.

However, not all substructural logics are decidable in this sense.
Most famously, the relevant logic R is not decidable.
This is partly because its proof theory is more complex than that of
other substructural logics. R differs from linear
logic and the Lambek calculus in having a straightforward treatment of
conjunction and disjunction. In particular, conjunction and disjunction
satisfy the rule of distribution:

p & (q
∨
r)
⊢
(p &
q)
∨
(p &
r)

The natural proof of distribution in any proof system uses both
weakening and contraction, so it is not available in the relevant logic
R, which does not contain weakening. As a result,
proof theories for R either contain distribution as a
primitive rule, or contain a second form of premise combination (so
called extensional combination, as opposed to the
intensional premise combination we have seen) which satisfies
weakening and contraction.

In recent years, a great deal of work has been done on the proof
theory of classical logic, inspired and informed by research
into substructural logics. Classical logic has the full complement of
structural rules, and is historically prior to more recent systems of
substructural logics. However, when it comes to attempting to
understand the deep structure of classical proof systems (and
in particular, when two derivations that differ in some superficial
syntactic way are really different ways to represent the one
underlying ‘proof’) it is enlightening to think of
classical logic as formed by a basic substructural logic, in which
extra structural rules are imposed as additions. In particular, it has
become clear that what distinguishes classical proof from its siblings
is the presence of the structural rules of contraction and weakening in
their complete generality (see, for example, Bellin et al.
2006 and the literature cited therein).

While the relevant logic R has a proof system more
complex than the substructural logics such as linear logic, which lack
distribution of (extensional) conjunction over disjunction, its
model theory is altogether more simple. A Routley-Meyer
model for the relevant logic R is comprised
of a set of pointsP with a three-place relation
R on P. A conditional A → B is
evaluated at a world as follows:

A → B is true at x if and
only if for each y and z where Rxyz, if
A is true at y, B is true at
z.

An argument is valid in a model just when in any point at
which the premises are true, so is the conclusion. The argument
A
⊢
B →
B is invalid because we may have a point x at which
A is true, but at which B → B is not.
We can have B → B fail to be true at x
simply by having Rxyz where B is true at y
but not at z.

The three place relation R follows closely the behaviour of
the mode of premise combination in the proof theory for a substructural
logic. For different logics, different conditions can be placed on
R. For example, if premise combination is commutative, we
place a symmetry condition on R like this:
Rxyz if and only if Ryxz. Ternary relational
semantics gives us great facility to model the behaviour of
substructural logics. (The extent of the correspondence between the
proof theory and algebra of substructural logics and the semantics is
charted in Dunn’s work on Gaggle Theory (1991) and is
summarised in Restall’s Introduction to Substructural
Logics (2000).)

Furthermore, if conjunction and disjunction satisfy the distribution
axiom mentioned in the previous section, they can be modelled
straightforwardly too: a conjunction is true at a point just when both
conjuncts are true at that point, and a disjunction is true at a point
just when at least one disjunct is true there. For logics, such as
linear logic, without the distribution axiom, the semantics
must be more complex, with a different clause for disjunction required
to invalidate the inference of distribution.

It is one thing to use a semantics as a formal device to model a
logic. It is another to use a semantics as an interpretive
device to apply a logic. The literature on substructural
logics provides us with a number of different ways that the ternary
relational semantics can be applied to describe the logical structure
of some phenomena in which the traditional structural rules do not
apply.

For logics like the Lambek calculus, the interpretation of the
semantics is straightforward. We can take the points to be linguistic
items, and the ternary relation to be the relation of concatenation
(Rxyz if and only if x concatenated with y
results in z). In these models, the structural rules of
contraction, weakening and permutation all fail, but premise
combination is associative.

The contemporary literature on linguistic classification extends the
basic Lambek Calculus with richer forms of combination, in which more
syntactic features can be modelled (see Moortgat 1995).

Another application of these models is in the treatment of the
semantics of function application. We can think of the points
in a model structure as both functions and data, and
hold that Rxyz if and only if x (considered as a
function) applied to y (considered as data) is z.
Traditional accounts of functions do not encourage this dual use, since
functions are taken to be of a ‘higher’ than their inputs
or outputs (on the traditional set-theoretic model of functions, a
function is the set of its input-output pairs, and
so, it can never take itself as an input, since sets cannot
contain themselves as members). However, systems of functions modelled
by the untyped λ-calculus, for example, allow for
self-application. Given this reading of points in a model, a point is
of type A → B just if whenever it takes inputs
of type A, it takes outputs of type B. The inference
rules of this system are then principles governing types of functions:
the sequent

(A → B) & (A →
C)
⊢
A →
(B & C)

tells us that whenever a function takes As to Bs and
As to Cs, then it takes As to things that
are both B and C.

This example gives us a model in which the appropriate substructural
logic is extremely weak. None of the usual structural rules
(not even associativity) are satisfied in this model. This example of a
ternary relational model is discussed in (Restall 2000, Chapter
11).

For the relevant logic R and its interpretation of
natural language conditionals, more work must be done in identifying
what features of reality the formal semantics models. This has been a
matter of some controversy, since not only is the ternary relation
unfamiliar to those whose exposure is primarily to modal logics with a
simpler binary accessibility relation between possible worlds,
but also because of the novelty of the treatment of negation
in models for relevant logics. It is not our place to discuss this
debate in much detail here, Some of this work is reported in the
article on
relevant logic
in this
Encyclopedia, and a book-length treatment of relevant logic in this
light is Mares’ Relevant Logic: a philosophical
interpretation (2004).

A comprehensive bibliography on relevant logic was put together by
Robert Wolff and can be found in Anderson, Belnap and Dunn 1992. The
bibliography in Restall 2000 (see
Other Internet Resources)
is not as comprehensive as Wolff’s, but it does
include material up to the present day.

Books on Substructural Logic and Introductions to the Field

Anderson, A.R., Belnap, N.D. Jr., and Dunn, J.M., 1992,
Entailment, Volume II, Princeton, Princeton University Press
[This book and the previous one summarise the work in relevant logic
in the Anderson–Belnap tradition. Some chapters in these books
have other authors, such as Robert K. Meyer and Alasdair
Urquhart.]

Schroeder-Heister, Peter, and Došen, Kosta, (eds), 1993,
Substructural Logics, Oxford University Press.
[An edited collection of essays on different topics in substructural
logics, from different traditions in the field.]

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